Liouville correspondence between the Short-Pulse Hierarchy and the Sine-Gordon Hierarchy

TL;DR

This paper establishes a Liouville correspondence linking the hierarchies of the Short-Pulse and Sine-Gordon equations through a transformation that relates their flows and conservation laws.

Contribution

It demonstrates that the transformation connecting the SP and SG equations extends to their entire hierarchies, revealing a deep structural correspondence.

Findings

The transformation relates the flows of the two hierarchies.

The Hamiltonian conservation laws correspond under the transformation.

The hierarchy structures are preserved between SP and SG equations.

Abstract

This paper considers the whole hierarchy of bi-Hamiltonian integrable equations associated to each of the Short-Pulse (SP) equation and the Sine-Gordon (SG) equation. We prove that the transformation that relates the SP equation with the SG equation also serves to establish the correspondence between their flows and Hamiltonian conservation laws in respective hierarchy.